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What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.

Thank you very much.

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See the accepted answer for this thread… – Martin Van der Linden Nov 2 '13 at 21:47
Nice..........+1 – Bhaskara-III Feb 10 at 11:46
@Bhaskara-III. Thank you! :) – Mark Feb 10 at 11:58
up vote 11 down vote accepted

If you think about the matrix as representing a linear transformation, then the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed. So for instance, the matrix $2I$ stretches a square of area 1 into a square with area 4, since the determinant is 4. This idea works in all dimensions too, not just 2 or 3!

This also translates well when you get in to more general mappings $f:\Bbb{R}^n\rightarrow\Bbb{R}^n$; if the function is nice enough, you can represent it "locally" by a linear transformation. The (absolute value of the) determinant of this linear transformation gives the "local" volume distortion of the function - i.e. how much the function is stretching or compressing regions of space near a point.

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